A Commutative Family of Integral Transformations and Basic Hypergeometric Series. I. Eigenfunctions

It is conjectured that a class of n-fold integral transformations {I(alpha)|alpha in {C}} forms a mutually commutative family, namely, we have I(alpha) I(beta)=I(beta) I(alpha) for all alpha, beta in {C}. The commutativity of I(alpha) for the two-fold integral case is proved by using several summation and transformation formulas for the basic hypergeometric series. An explicit formula for the complete system of the eigenfunctions for n=3 is conjectured. In this formula and in a partial result for n=4, it is observed that all the eigenfunctions do not depend on the spectral parameter alpha of I(alpha).Comment: Basic parameters are replaced to make the notation consistent with the standard Macdonald polynomials: q is replaced by t, and p^{1/2} is replaced by

A trial to find an elliptic quantum algebra for sl2sl_2 using the Heisenberg and Clifford algebra

A Heisenberg-Clifford realization of a deformed $U(sl_{2})$ by two parameters $p$ and $q$ is discussed. The commutation relations for this deformed algebra have interesting connection with the theta functions.Comment: 4 page

Kernel Functions for Difference Operators of Ruijsenaars Type and Their Applications

A unified approach is given to kernel functions which intertwine Ruijsenaars difference operators of type A and of type BC. As an application of the trigonometric cases, new explicit formulas for Koornwinder polynomials attached to single columns and single rows are derived.Comment: 40 pages. Three sections are added in Appendix, as well as several comments and reference